The time between calls is exponentially distributed with a...

The time between calls is exponentially distributed with a mean time between calls of 10 minutes. (a) What is the probability that the time until the first call is less than 5 minutes? (b) What is the probability that the time until the first call is between 5 and 15 minutes? (c) Determine the length of an interval of time such that the probability of at least one call in the interval is $0.90 .$ (d) If there has not been a call in 10 minutes, what is the probability that the time until the next call is less than 5 minutes? (e) What is the probability that there are no calls in the intervals from 10: 00 to $10: 05,$ from 11: 30 to $11: 35,$ and from 2: 00 to $2: 05 ?$ (f) What is the probability that the time until the third call is greater than 30 minutes? (g) What is the mean time until the fifth call?

The time between calls is exponentially distributed with a mean time between calls of 10 minutes.
(a) What is the probability that the time until the first call is less than 5 minutes?
(b) What is the probability that the time until the first call is between 5 and 15 minutes?
(c) Determine the length of an interval of time such that the probability of at least one call in the interval is $0.90 .$
(d) If there has not been a call in 10 minutes, what is the probability that the time until the next call is less than 5 minutes?
(e) What is the probability that there are no calls in the intervals from 10: 00 to $10: 05,$ from 11: 30 to $11: 35,$ and from 2: 00 to $2: 05 ?$
(f) What is the probability that the time until the third call is greater than 30 minutes?
(g) What is the mean time until the fifth call?

Key Concepts

Gamma/Erlang Distribution

The Gamma distribution, and specifically its discrete version known as the Erlang distribution, generalizes the exponential distribution to model the waiting time until the kth event in a Poisson process occurs. When k is an integer, the Erlang distribution is used; its mean is k times the mean of the exponential distribution, making it useful for analyzing scenarios that involve multiple events.

Poisson Process

A Poisson process is a statistical process that models events occurring randomly over time. The exponential distribution describes the time between successive events in this process, and its properties are used to calculate probabilities of event occurrences over specified intervals. In a Poisson process, the number of events occurring in non-overlapping intervals is independent.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by the rate parameter (?), which is the reciprocal of the mean. Its probability density function is defined as f(t) = ? exp(??t) for t ? 0, and it is widely used in reliability analysis, queueing theory, and survival analysis.

Memoryless Property

A key characteristic of the exponential distribution is its memoryless property, which means that the probability of an event occurring in the next interval does not depend on how much time has already elapsed. This property is unique to the exponential distribution among continuous distributions and is analogous to the geometric distribution in discrete settings.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the exponential distribution gives the probability that the waiting time until an event occurs is less than or equal to a certain value. It is expressed as F(t) = 1 ? exp(??t). The CDF is integral in problems that ask for the probability that an event happens within or beyond a specified time period.

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